A non-stationary model of the incompressible viscoelastic Kelvin-Voigt fluid of non-zero order in the magnetic field of the Earth Текст научной статьи по специальности «Математика»

MSC 35G61

DOI: 10.14529/ mmp190304

A NON-STATIONARY MODEL OF THE INCOMPRESSIBLE VISCOELASTIC KELVIN-VOIGT FLUID OF NON-ZERO ORDER IN THE MAGNETIC FIELD OF THE EARTH

A.O. Kondyukov1, T.G. Sukacheva1'2

1 Novgorod State University, Velikiy Novgorod, Russian Federation 2South Ural State University, Chelyabinsk, Russian Federation E-mails: [email protected], [email protected]

We investigate the Cauchy-Dirichlet problem for a system of Oskolkov equations of nonzero order. The considered mathematical model describes the flow of an incompressible viscoelastic Kelvin-Voigt fluid in the magnetic field of the Earth. The model takes into account that the fluid is subject to various external influences, which depend on both the coordinate of the point in space and the time. The first part of the paper presents the known results obtained by the authors earlier and based on the theory of solvability of the Cauchy problem for semilinear nonautonomous Sobolev type equations. In the second part, we reduce the considered mathematical model to an abstract Cauchy problem. In the third part, we prove the main result that is the theorem on the existence and uniqueness of the solution. Also, we establish the conditions for the existence of quasi-stationary semitrajectories, and describe the extended phase space of the model under study. In this paper, we summarize our results for the Oskolkov system that simulates the motion of a viscoelastic incompressible Kelvin-Voigt fluid of zero order in the magnetic field of the Earth.

Keywords: magnetohydrodynamics; Sobolev type equations; extended phase space; incompressible viscoelastic fluid.

Introduction

The Oskolkov's system of equations

K 1_ . 1

(1 -xV2)vt = uV2v-(v ■ /3iV2wi--Vp-2Q x v-\--(V x b) x b+f \

P PV

V •v = 0, V- b = 0, bt = 8V2b + Vx (v x b) + f2, (1)

-^-=v + alwl, a,GK_, A G R+, l = T7K,

dt

simulates the flow of an incompressible viscoelastic Kelvin-Voigt fluid [1] of nonzero order K in the magnetic field of the Earth. Here the vector functions v = (vi(x,t),v2(x,t),... ,vn(x,t)) and b = (b1(x,t),b2(x,t),... ,bn(x,t)) characterize fluid velocity and magnetic induction, respectively, p = p(x, t) is the pressure, k is the coefficient of elasticity, v is the coefficient of viscosity, Q is the corner velocity, 8 is the magnetic viscosity, v is the magnetic permeability, p is the density, and the parameters Pi, I = 1, K determine the time of pressure retardation (delay). The absolute terms f1 = f,..., fn), fl = fl(x,t),f2 = f2(x,t) correspond to external influences on the fluid.

Consider the first initial boundary value problem for system (1): v(x, 0) = v0(x), b(x, 0) = b0(x), wi(x, 0) = wio(x) x G D,

(2)

v{x, t) = 0, b{x, t) = 0, Wi{x, t) = 0 (x, t) G dD x R+, I = 1, K

under assumption that ^ =1 and p =1. Here D C Rn is a bounded domain with the boundary dD of the class Cr.

Problems that are similar to problem (1), (2) take place, for example, in mathematical modelling in geophysical sciences [2].

Note that degenerate models of magnetohydrodynamics were previously studied by the authors in the papers [3-6]. A distinctive feature of the present paper is the presence of the vector-functions f1 = (fl,---,f^), f1 = fl(x,t),f2 = f2(x,t) in the right hand side of equation (1). The paper [7] considers the model of magnetohydrodynamics, which takes into account various external influences for K = 0. The case of K > 0 is investigated for the first time.

Problem (1), (2) is investigated in the framework of the theory of semilinear Sobolev type equations [8,9]. The main tool of the study is the notion of a relative p-sectorial operator and a resolving degenerate semigroup of operators generated by this operator [10,11]. We prove the theorem on the existence and uniqueness of the solution to this problem, and describe the extended phase space of the problem.

The article consists of three sections. Section 1 gives the known necessary results of the theory of semi-linear Sobolev type equations [10,12]. Section 2 reduces problem (1),

(2) to the Cauchy problem for the semi-linear Sobolev type equation. Section 3 presents the theorem on the existence and uniqueness of the solution to the considered problem, shows that the solution is a quasi-stationary trajectory, and describes the extended phase space of the problem.

1. Semi-Linear Non-Stationary Sobolev Type Equations

Let U and F be Banach spaces, the operator L G L(U; F), i.e. L is linear and continuous, the operator M : dom M ^ F be linear, closed and densely defined in U, i.e. M G Cl(U; F). Denote UM = {u G dom M : ||u|| = \\Mu\\F + ||u||w}. Let the operator F G Cr(UM; F). We suppose that the operator F G Cr(UM; F), and the function f GCr(R +; F).

Consider the Cauchy problem

u(0) = u0 (3)

for the semi-linear non-stationary Sobolev type equation

Lu = Mu + F (u) + f (t). (4)

By a local solution (hereinafter, solution) to problem (3), (4) we mean the vector function u G Cr ((0, T); UM) that satisfies equation (4) and such that u(t) ^ u0 for t ^ 0+.

Let the operator M be strongly (L,p)-sectorial [12]. It is well known that, under this condition, problem (3), (4) can have several solutions [13]. Therefore, we are interested in only such solutions to problem (3), (4) that are quasi-stationary semitrajectories.

Definition 1. Suppose that the space U splits into the direct sum U = U0 © U1 such that ker L C U0. A solution u = v + w to equation (4), where v(t) G U0 and w(t) G U1 for all t G (0,T), is called a quasi-stationary semitrajectory, if LV = 0.

Also, it is known [12] that problem (3), (4) can has no solutions for some u0 GUM. Therefore, we introduce another definition.

Definition 2. The set B C Um x R + is called the extended phase space of equation (4), if for any point u0 G UM such that (u0, 0) G B0 there exists a unique solution to problem,

(3), (4), and (u(t),t) G Bl.

We consider problem (3), (4) under the condition that the operator M is strongly (L,p)-sectorial [12]. In this case, the problem can have no solution for some u0 G UM, and even if there exists a solution for all u0 G UM, then the solution can be non-unique.

It is well known that if the operator M is strongly (L,p)-sectorial, then U = U0 ©U1, F = F0©F1, where U0 = (p GU : UV = 0 3t G R+}, F0 = gF : FV = 0 3t G R+} are the kernels, and U1 = (u G U : limt^0+ Utu = u},F1 = (f G F : limt^0+ Ftf = f} are the images of the analytic solving semigroups

ut=i~I pt=bi\ ^

r r

of the linear homogeneous equation

Lu = Mu, (6)

where r C SQ a(M) is a contour such that arg ^ ^ ±6 for ^

Denote by Lk and Mk the restrictions of the operators L and M on Uk (Uk fl dom M), k = 0, 1, respectively. Then Lk : Uk ^ Fk, Mk : Uk f dom M ^ Fk, k = 0, 1, and the restrictions M0 and L1 of the operators M and L on the spaces U0 f dom M and U1 are linear continuous operators and have bounded inverse operators.

Therefore, problem (3), (4) is reduced to an equivalent system, which we call the normal form of problem (3), (4):

Ru0 = u0 + G(u) + g(t), u0(0) = u0, (7)

u1 = Su1 + H (u) + h(t) u1(0) = u1, (7)

where uk G Uk, k = 0,1, u = u0 + u1, the operators R = M-1 L0, S = L-1Mb G = M0-1 (I - Q)F, H = L^QF, g = M-1 (I - Q)f, h = L-1 Qf.

Here Q G L(F)(= L(F; F)) is the projector that splits the space F as required. Further, we study only the quasi-stationary semitrajectories of equation (4), for which Ru0 = 0. To this end, we assume that the operator R is bi-splitting , i.e. the kernel ker R and the image im R are completed in the space U. Suppose that U00 = ker R. Denote by U01 = U0 ©U00 a complement of the subspace U00. Then the first equation of normal form (7) is reduced to m nn m „ s „ s

V ^ Ru01 = u00 + u01 + G(u) + g(t), (8)

where u = u00 + u01 + u1.

Theorem 1. Let the operator M be strongly (L,p)-sectorial, and the operator R be bisplitting. Suppose that there exists the quasi-stationary semitrajectory u = u(t) of equation (4). Then u = u(t) satisfies the following relations:

0 = u00 + u01 + G(u) + g (t),u01 = const. (9)

It is known that if the operator M is strongly (L,p)-sectorial, then the operator S is sectorial. Therefore, on U1, the operator S generates an analytic semigroup, which we denote by (Uf : t > 0}, since the operator Uf is a restriction of the operator Ut on U1.

Since U = U0 ©U1, then there exists the projector P G L(U) corresponding to this splitting. It is easy to see that P G L(UM). Then the space UM splits into the direct sum UM = UM ©U1 such that the embedding UM C Uk, k = 0, 1, is dense and continuous. Further, denote by AVv the Frechet derivative at the point v G V of the operator A defined on the Banach space V.

Theorem 2. Let the operator M be strongly (L,p)-sectorial, the operator R be bi-splitting, the operator F G C^(UM; F), and the vector-function f G C^(R+; F). Suppose that the following conditions are fulfilled.

(i) In the neighborhood Ou0 C UM of the point u0, the following relation takes place:

0 = uO1 + (I - Pr)(G(u00 + uO1 + u1) + g(t)). (10)

(ii) The projector PR G L(UM), and the operator I + PRG'uo ■ UM ^ UMO is the topological linear isomorphism (UM = UM RU00).

(iii) For the analytic semigroup {Uf ■ t > 0}, the following condition is fulfilled:

T

/||U11|1;uM)dt< M Vt G R+. (11)

0

Then there exists the unique solution to problem (3), (4), which is the quasi-stationary semitrajectory.

Remark 1. Condition (11) is not satisfied for ordinary analytic semigroups having the estimate ||Uf||L(Ui;Um) < const/t. Denote by U^ = [U1; UM}a, a G [0,1], some interpolation space constructed by the operator S. Complete the condition F G C^ (UM; F) of Theorem 2 with the condition H G C^ (UM; UV), and replace Condition (11) with

T

Jnu{||L(U 1 u) dt< m, T G R+. (12)

0

Then the statement of Theorem 2 is the same.

Let Uk and Fk be Banach spaces, the operators Ak G L(Uk, Fk), and the operators Bk ■ dom Bk ^ F be linear and closed with domain of definitions dom Bk, which are dense in Uk, k = 1, 2. Construct the spaces U = U1 x U2, F = F1 x F2 and the operators L = A1 ® A2, M = B1 ® B2. By the construction, the operator L G L(U; F), and the operator M ■ dom M ^ F is linear, closed and densely defined, dom M = dom B1 x dom B2.

Theorem 3. Let the operators Bk be strongly (Ak,pk)-sectorial, k = 1, 2, and p1 > p2. Then the operator M is strongly (L,p1 )-sectorial.

2. Reduction to Abstract Cauchy Problem

In order to reduce problem (1), (2) to problem (3), (4), we transfer from system (1) to the system

K

(1 - xV2)vt = vV2v - (v • V)v + ß,V2w, - P - x v + (V x b) x b + f \

1=1

V(V • v) = 0, V(V • b) = 0, bt = SV2b + Vx (v x b) + f2. (13)

dm.

= v + aiwi, a, G R_, ß, G R+, l = 1,K.

dt

We are interested in solvability of problem (13), (2). Following the

paper [12], we introduce the spaces H2, H2K, H2, and Hn. Here H2 and

H2 are subspaces of the solenoid functions in the spaces (W2(D))n f

o

f (W\(D))n and (L2(D))n, respectively, and H2n and Hf are their orthogonal (in the sense of (L2(D))n) complements. Denote by £ both the orthoprojector on H2 and its

restriction on the space (W22(D))n if (W\(D))n. Suppose that n = I — E. The equality A = V2En : H2 © Hi ^ H2 © Hn, where En is a unit matrix of order n, defines a linear continuous matrix operator with discrete finite-multiple spectrum a (A) C R that tends only to —to. The formula Bv : v ^ V(V-v)(Bb : b ^ V(V-b)) gives the linear continuous surjective operator Bv (Bb) : H2 © H ^ Hn with the kernel ker Bv = Bb = H2. We use the natural isomorphism of the direct sum and the Cartesian product of Banach spaces in order to introduce the spaces Ui0 = H2 x H^ x Hp , Fi0 = H2 x Hn x Hp, where

Hp = H^ ; Uu = H2 n H1 = H2 x H2 , and Tu = L2 = HaxH,,! = l, K. Then spaces Ui = ©K=0 Uii , Fi = ©K=0 F

p — AA^ , — AA I I AA — AAct 11,

il.

The operators Ai and Bi are defined by the formulas Ai = diag A , Ek] , where

A , a4i ^ a = ( E(I — AA)E EA(I — AA)n

O Or i U(I — AA)E nA(I — AA)n

2

Bi = (Bj )ij=i , where

/ vEA vEA O \ / ft EA ... pK EA

Bi i ^ vnA vnA —I , Bi 2 ^ pinA ... pKnA \ O B O ) \ O ... O

In the matrix Bi i, B = V(V ■ v) — V(V ■ b) = Bv — Bb. The matrix B2 i contains K rows of the form (I, I, O), B 22 = diag [a i}..., aK ].

Remark 2. Denote by A2 the restriction of the operator EA on H2. According to the Solonnikov-Vorovich-Yudovich theorem, the spectrum a(A2) is real, discrete, finite-multiple, and tends only to —to.

Theorem 4.

i) The operators Ai; Bi belong to L(U; f;l), and if k-i / a(A), then the operator Ai is bi-splitting, ker Ai = {0} x {0} x Hp x {0} x ... x {0}, im Ai = H2 x Hn x {0} x^x

K

F2 x ... x fk .

ii) If A-i / a(A) U a(A2), then the operator Bi is (Ai, 1)-bounded.

Proof. The statement of the theorem is the direct corrolary of the results obtained in [12].

□

Remark 3. The (L,p)-bounded operator is defined, for example, in [12].

Suppose that U2 = F2 = L2(D). The equality B2 = 5V2 : def B2 ^ F2 defines the

o

linear closed and densely defined operator B2, domB2 = W22(D) f W2(D). Let A2 = I. Theorem 5. The operator B2 is strongly A2-sectorial.

Proof. The statement of the theorem follows from the sectoriality of the operator B2 [14].

□

Let U = U x U2, F = F x F2.

The vector u of the space U has the form u =col(u2,un,up,wi,...,wK,ub), where col(u2, un, up,w-]_,..., wK) £ U, and ub £ U2, ub = (b2, bn), b2 £ H2, bn £ H. The vector f £ F has the similar form. Define the operators L and M by the equalities L = Ai © A2 and M = Bi © B2. The operator L belongs to L(U; F), and the operator M : dom M ^ F is linear, closed and densely defined, dom M = U x dom B2.

o

Theorem 6. Let k 1 £ a(A), then the operator M is strongly (L, 1)-sectorial.

Proof. By virtue of Theorem 4 and the results of Paragraph 3.1. [12], the operator B1 is strongly (A1, 1)-sectorial. Therefore, taking into account Theorems 3 and 5, we obtain that the statement of the theorem is true. ^

Let us construct the nonlinear operator F. Represent the operator as

F = Fi 0 F2,

where

Fi = Fi(ua ,un ,b) = col(-E(((ua + un) •V)(uia + Un) - 2Q x (ua + un) + (V x b) x b + f1), -Tl(((ua + un) ■ V)(ua + un) -2Qx (ua + ii,) + (Vx6)xfe + /1),01_^0),

K+1

F2 = F2(ua, un ,b) = V x ((ua + un) x b) + f2. In our case, UM = U1 x dom B2, since the operator B1 is continuous.

Theorem 7. The operator F belongs to Crx(UM; F).

Proof. The statement of the theorem follows from the fact that for any u £ UM the operator FU belongs to L(UM; F), the second Frechet derivative F"u of the operator F is the continuous bilinear operator that belongs to UM x UM in F, and F"' = O (similarly to [12]). n

Therefore, we have reduced (1), (2) to (3), (4), and we can consider these two problems to be equivalent. Let us verify the conditions of Theorems 1 and 2.

3. Theorem on Existence and Uniqueness of Solution

By virtue of Theorem 6 and the results of Paragraph 3.1. [12], there exists the analytic semigroup {U1: t£R+} of the resolving operators of equation (6). In this case, U1 is naturally represented as Ul = V1 x Wl, where Vt(Wt) is the restriction of the operator Ul on U1 (U2). Since B2 is sectorial, then Wt = exp(tB2), and, therefore, the kernel of this semigroup is W° = {0}, and the image of this semigroup is W1 = U2.

Consider the semigroup {Vt : t £ R+}. By virtue of Theorems 4 and 6 and results of Paragraph 3.1. [12], this semigroup is extended to the group {Vt : t £ R}. The kernel of the semigroup is V0 = U00 ©W?1, where U00 = {0} x {0} x Hp x {0} x ... x {0}(= ker A1 due to Theorem 5), and W01 = £A-1A-n[Hn] x H x {0} x ... x {0} . Here Ak = I - kA,

S-V-'

k+1

AKn is the restriction of the operator nA-1 on Hn. It is known that if k-1 / a (A) U a(Aa), then the operator AKn : Hn ^ H^ is topological linear isomorphism [12]. Denote by W1 the image of V1. Then, since the operator B1 is strongly (A1,1)-sectorial, then the space W1 decomposes into the direct sum of the subspaces W1 = W°0 © W1 © U\.

Construct the operator R (see (5), (6)). In our case, R = B-,1 A10 £ L(W100 ©W^1), where A10(B10) is the restriction of the operator A1(B1) on W0 © W1. Note that the operator B—1 exists due to Theorem 6 and the corresponding results obtained in [12]. By construction, ker R = W0, and the paper [15] shows that im R = W1. Therefore, the operator R is bi-splitting. Denote by PR the projector of the space W0 © W1 on W0 along W1. Taking into account the structure of the space UM, we obtain that the projector Pr belongs to L(U°M), where U°M = Um n (W00 ©W01 )(= W^0 ©W^1). Therefore, the following lemma is valid.

Lemma 1. Suppose that k 1 / o(A) U a(Aa). Then the operator R is bi-splitting, and PRe ).

Consider the projectors

Pk = diag [Pk, 0], Qk = diag[Qk, 0], k = 0, 1,

see [12] for a detailed description of these projectors. Taking into account the results of [12] and the fact that the kernel W0 = {0}, we obtain that I - P = (P0 + Pi) x O, Q = (I - Qo - Qi) x I, P : U ^ U 1, Q : F ^ F 1.) Then, apply the projector IP to equation (4) in our situation and obtain the equations

K

' RV2w. — u __ OH x (u + u ) +

(14)

(V x b) x b + f 1(t)) = 0, Bun = 0, Bbn = 0. Hence, by virtue of Theorem 1 and the properties of the operator B, we obtain the necessary condition for existence of the quasi-stationary trajectory un = 0, bn = 0, i.e. all solutions to problem (2), (13) (if they exist) necessarily belong to the plane B = {u G Um : Un = 0, bn = 0}.

Since Hup = up, we obtain relation (9) from the first equation of (14), i.e. in our case,

K

up = n(vAua - (ua • V)ua + RV2wi - 2H x ua + (V x ba) x ba + f 1(t)). (15)

1=1

Lemma 2. Under the conditions of Lemma 1, any solution to problem (1), (2) belongs to the set

n(vA(ua + un) - ((ua + un) • V)(uCT + un) + ^ RiV2wi - up - 2H x (ua + un) +

l=1

M = {u G Um : un = 0, bn = 0, up = n(vAa - (ua • V)ua + ^ RiV2w - 2Q x +

+(Vx bCT) x ba) + f 1(t)}. 1=1 Remark 4. Relation (15) gives condition A2) of Theorem 2 for any point u0 G UM(= U00 x {0}). Therefore, similarly to [12], we obtain that the set M is a simple Banach manifold that is C^-diffeomorphic to the subspace U1 x U2, and can be the extended phase space of problem (1), (2) ((13), (2)).

o

Let us verify conditions (11), (12). Construct the space Ua = U1 x W2(D). Obviously, this space is the interpolation space for the pair [U, UM]a, and a = 1/2. As noted above, the semigroup {Ul : t G R+} is extended to the group {V/ : t G R} on UI, where V/ is the restriction of the operator V1 on U1 Since U^ = UM fl U1 by construction, the operator B1 is continuous by virtue of Theorem 4, and the semigroup {U1 : t G R+} is uniformly bounded, we obtain the inequality

T T

J iiV/iil^I;^) dt < const x ||B1||mF^J \\Vl\\c(ul) dt < to, t g R+. (16)

According to Sobolev's inequality [12], the semigroup {W1 : t G R +} satisfies the estimate T

i WW1 \l °1 , ,,dt< to. (17)

/ 11 L(dom B2 ;W2(D)) V '

0

11

Suppose that U^ = Ua fU1, where U1 = U1 x U2. Then inequalities (16) and (17) give the following lemma.

Lemma 3. Under the conditions of Lemma 1, relation (11) takes place.

Taking into account condition (12), we obtain the operator H as follows. The operator H is naturally represented as H = H10H2, where H1 = A-1 (I — Q0 — Q1)F1, and H2 = F2 (A11 is the restriction of the operatorA1 on U1). For the operator H, there is the statement that is similar to Theorem 7 for the operator F, i.e. H £ C^(UM; U^), where U^ = Ua nW1.

Therefore, all the conditions of Theorem 2 are satisfied. Therefore, the following statement is valid.

Theorem 8. Suppose that k-1 / a(A) U a(Aa). Then for any u0 such that u0 £ M, and some T £ R+, there exists the unique solution u = (ua, 0,up,ub) to problem (1), (2), which is a quasi-stationary trajectory, and u(t) £ M for all t £ (0,T).

Acknowledgements. The authors express their gratitude to Professor G.A. Sviridyuk for his attention and constructive criticism. The work was supported by the Government of the Russian Federation (act 211, contract no. 02.A03.21.0011).

References

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2. Hide R. On Planetary Atmospheres and Interiors. Providence, American Mathematical Society, 1971.

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Received April 21, 2019

УДК 517.9 Б01: 10.14529/шшр190304

НЕСТАЦИОНАРНАЯ МОДЕЛЬ НЕСЖИМАЕМОЙ ВЯЗКОУПРУГОЙ ЖИДКОСТИ КЕЛЬВИНА - ФОЙГТА НЕНУЛЕВОГО ПОРЯДКА В МАГНИТНОМ ПОЛЕ ЗЕМЛИ

А.О. Кондюков1, Т.Г. Сукачева,12

1 Новгородский государственный университет им. Ярослава Мудрого, г. Великий Новгород, Российская Федерация 2Южно-Уральский государственный университет, г. Челябинск, Российская Федерация

В работе исследуется задача Коши - Дирихле для системы уравнений Осколко-ва ненулевого порядка. Рассматриваемая математическая модель описывает течение несжимаемой вязкоупругой жидкости Кельвина - Фойгта в магнитном поле Земли. При этом учитывается, что на жидкость оказывают влияние различные внешние воздействия, зависящие как от координаты точки в пространстве, так и от времени. В первой части работы излагаются известные результаты, полученные авторами ранее, из теории разрешимости задачи Коши для полулинейных неавтономных уравнений соболевского типа. Во второй части проводится редукция рассматриваемой математической модели к указанной абстрактной задаче Коши. В третьей части доказывается основной результат - теорема существования и единственности решения. Находятся условия существования квазистационарных полутраекторий, а также описывается расширенное фазовое пространство исследуемой модели. Представленные в статье исследования обобщают результаты авторов для системы Осколкова, моделирующей движение вязкоупругой несжимаемой жидкости Кельвина - Фойгта нулевого порядка в магнитном поле Земли.

Ключевые слова: магнитогидродинамика; уравнения соболевского типа; расширенное фазовое пространство; несжимаемая вязкоупругая жидкость.

Литература

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Алексей Олегович Кондюков, кандидат физико-математических наук, старший преподаватель, кафедра алгебры и геометрии, Новгородский государственный университет им. Ярослава Мудрого (г. Великий Новгород, Российская Федерация), [email protected].

Тамара Геннадьевна Сукачева, доктор физико-математических наук, профессор, кафедра алгебры и геометрии, Новгородский государственный университет им. Ярослава Мудрого (г. Великий Новгород, Российская Федерация); научно-исследовательская лаборатория «Неклассические уравнения математической физики:», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация) , [email protected].

Поступила в редакцию 21 апреля 2019 г.